As in the homogeneous context, there are profound and interesting links
to branching processes (in random environment). In [1]}, in the
setting of discrete space, invariance principles have been derived for
the position of the front of the PAM as well as the
position of the maximum of BRWRE. Furthermore, it has been shown that the
distance between these two quantities is in \(O(\ln t)\)  as \(t \rightarrow \infty .\) 
In this context, a subtle but important difference to the homogeneous
setting is that the solution to (REF ) and the maximum of
branching Brownian motion in random environment (BBMRE; see Section
REF  below for the precise definition) do exhibit a slightly
more involved interrelation. In particular, neither can we directly
transfer the sub-sequential tightness result of Kriechbaum [2]}
for the law of the maximum of branching random walk in random environment
(BRWRE) in the context of [1]} to the setting of
(REF ), nor can we directly obtain a respective non-tightness
result for BBMRE from our unbounded transition fronts for the solution to
(REF ).
Furthermore, it is trivial that the distribution function
\(w^{\xi \equiv \text{const}}(t,\cdot )\)  of the maximum of a BBMRE at time
\(t,\)  which is the solution to (REF ) with \(\xi \equiv \text{const}\) ,
is non-increasing in space. This again is in stark contrast to
Theorem REF , which states that this is not the case
for the solution to (REF ) anymore if \(\xi \)  exhibits ‘‘enough”
irregularity.
